Integral exotic sheaves and the modular Lusztig-Vogan bijection

TL;DR

This paper proves that the Lusztig-Vogan bijection, linking dominant weights to equivariant vector bundles on nilpotent orbits, is characteristic-independent, enabling extension of characteristic-0 properties to positive characteristic cases.

Contribution

It establishes the characteristic-independence of the Lusztig-Vogan bijection, facilitating the transfer of known properties from characteristic zero to positive characteristic.

Findings

Lusztig-Vogan bijection is independent of the characteristic of the field.

Extension of properties from characteristic zero to positive characteristic.

Progress towards proving the Humphreys conjecture for reductive groups.

Abstract

Let G be a reductive group over an algebraically closed field k of very good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for G and the set of irreducible G-equivariant vector bundles on nilpotent orbits, conjectured by Lusztig and Vogan independently, and constructed in full generality by Bezrukavnikov. In characteristic 0, this bijection is related to the theory of 2-sided cells in the affine Weyl group, and plays a key role in the proof of the Humphreys conjecture on support varieties of tilting modules for quantum groups at a root of unity. In this paper, we prove that the Lusztig-Vogan bijection is (in a way made precise in the body of the paper) independent of the characteristic of k. This allows us to extend all of its known properties from the characteristic-0 setting to the general case. We also expect this result to be a step towards a proof of the Humphreys conjecture on support varieties of tilting modules for reductive groups in positive characteristic.